Necessary and sufficient conditions for the existence of stabilizing feedback for control systems

ẋ = f (t, x) + g(t, x)u (1.1) x ∈ Rn, t 0, u ∈ U where f (t, x) and g(t, x) are C0 mappings on R+ × Rn , locally Lipschitz with respect to x ∈ Rn , with f (t, 0) = 0 for all t 0 and U ⊆ Rm is a convex set that contains 0 ∈ Rm . Our objective is to give necessary and sufficient conditions for the existence of a C0 function k : R+ × Rn → U , with k(·, 0) = 0, k(t, x) being locally Lipschitz with respect to x ∈ Rn , such that 0 ∈ Rn is globally asymptotically stable (GAS) for the closed-loop system (1.1) with